Splines over triangulations and splines over quadrangulations (tensor product splines) are two common ways to extend bivariate polynomials to splines. However, combination of both approaches leads to ...splines defined over mixed triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes are especially useful for representing complicated geometries obtained e.g. from trimming. As (bi-)linearly parameterized mesh elements are not flexible enough to cover smooth domains, we focus in this work on the case of planar mixed meshes parameterized by (bi-)quadratic geometry mappings. In particular we study in detail the space of C1-smooth isogeometric spline functions of general polynomial degree over two such mixed mesh elements. We present the theoretical framework to analyze the smoothness conditions over the common interface for all possible configurations of mesh elements. This comprises the investigation of the dimension as well as the construction of a basis of the corresponding C1-smooth isogeometric spline space over the domain described by two elements. Several examples of interest are presented in detail.
•We study C1-smooth isogeometric spline functions of general degree over interfaces between triangles and quadrilaterals.•We focus mostly on the case of mesh elements parameterized by (bi-)quadratic geometry mappings.•We present a framework to analyze the smoothness conditions over the interface for all possible configurations of elements.•We investigate the dimension and construct a basis of the corresponding C1-smooth isogeometric spline space.
In this paper we introduce a C1 spline space over mixed meshes composed of triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis. In this context, a mesh is considered to be a ...partition of a planar polygonal domain into triangles and/or quadrilaterals. The proposed space combines the Argyris triangle, cf. Argyris et al. (1968), with the C1 quadrilateral element introduced in Brenner and Sung (2005), Kapl et al. (2020) for polynomial degrees p≥5. The space is assumed to be C2 at all vertices and C1 across edges, and the splines are uniquely determined by C2-data at the vertices, values and normal derivatives at chosen points on the edges, and values at some additional points in the interior of the elements.
The motivation for combining the Argyris triangle element with a recent C1 quadrilateral construction, inspired by isogeometric analysis, is two-fold: on one hand, the ability to connect triangle and quadrilateral finite elements in a C1 fashion is non-trivial and of theoretical interest. We provide not only approximation error bounds but also numerical tests verifying the results. On the other hand, the construction facilitates the meshing process by allowing more flexibility while remaining C1 everywhere. This is for instance relevant when trimming of tensor-product B-splines is performed.
In the presented construction we assume to have (bi)linear element mappings and piecewise polynomial function spaces of arbitrary degree p≥5. The basis is simple to implement and the obtained results are optimal with respect to the mesh size for L∞, L2 as well as Sobolev norms H1 and H2.
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